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Kusner's Conjecture
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called " metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of a d-dimensional Euclidean space is d+1, achieved by a regular simplex, and the equilateral dimension of a d-dimensional vector space with the Chebyshev distance (L^\infty norm) is 2^d, achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance (L^1 norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d, achieved by a cross polytope. Lebesgue spaces The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the L^p norm \ \, x\, _p=\left(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\right)^. The equilateral dimension o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertex corners. The tetrahedron is the simplest of all the ordinary convex polytope, convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean geometry, Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid (geometry), pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archiv Der Mathematik
'' Archiv der Mathematik'' is a peer-reviewed mathematics journal published by Springer, established in 1948. Abstracting and indexing The journal is abstracted and indexed in: Springer. 2022 * * * * According to the '' |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External links ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric And Functional Analysis
''Geometric and Functional Analysis'' (''GAFA'') is a mathematical journal published by Birkhäuser, an independent division of Springer-Verlag. The journal is published approximately bi-monthly. The journal publishes papers on broad range of topics in geometry and analysis including geometric analysis, riemannian geometry, symplectic geometry, geometric group theory, non-commutative geometry, automorphic forms and analytic number theory, and others. ''GAFA'' is both an acronym and a part of the official full name of the journal. History ''GAFA'' was founded in 1991 by Mikhail Gromov and Vitali Milman. The idea for the journal was inspired by the long-running Israeli seminar series "Geometric Aspects of Functional Analysis" of which Vitali Milman had been one of the main organizers in the previous years. The journal retained the same acronym as the series to stress the connection between the two. Journal information The journal is reviewed cover-to-cover in Mathematical Revi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, lengt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dvoretzky's Theorem
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids. A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic geometric analysis (also called ''asymptotic functional analysis'' or the ''local theory of Banach spaces''). Original formulations For every natural number ''k'' ∈ N and every ''ε'' > 0 there exists a natural number ''N''(''k'', ''ε'') ∈ N such that if (''X'', ‖·‖) is any normed space of dimension ''N''(''k'', ''ε''), t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Mazur Compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q(n) of n-dimensional normed spaces. With this distance, the set of isometry classes of n-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum. Definitions If X and Y are two finite-dimensional normed spaces with the same dimension, let \operatorname(X, Y) denote the collection of all linear isomorphisms T : X \to Y. Denote by \, T\, the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between X and Y is defined by \delta(X, Y) = \log \Bigl( \inf \left\ \Bigr). We have \delta(X, Y) = 0 if and only if the spaces X and Y are isometrically isomorphic. Equipped with the metric ''δ'', the space of isometry classes of n-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum. Many authors prefer to work with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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